Retrieving "Complete Manifold" from the archives

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1. Riemannian Manifold

   Linked via "complete"

   $$L(\gamma) = \int_a^b \sqrt{g(\dot{\gamma}(t), \dot{\gamma}(t))} dt$$
   If the manifold is complete (meaning all geodesics defined for all time remain defined), then any two points $p$ and $q$ are connected by at least one minimizing geodesic, establishing the metric space structure intrinsic to the Riemannian manifold. In spaces of constant positive curvature, such as those studied in [elliptic geometry](/entries/elliptic-geo…